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Neighbor sum distinguishing total chromatic number of planar graphs

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  • Xu, Changqing
  • Li, Jianguo
  • Ge, Shan

Abstract

Let G = (V(G), E(G)) be a graph and ϕ be a proper k-total coloring of G. Set fϕ(v)=∑uv∈E(G)ϕ(uv)+ϕ(v), for each v ∈ V(G). If fϕ(u) ≠ fϕ(v) for each edge uv ∈ E(G), the coloring ϕ is called a k-neighbor sum distinguishing total coloring of G. The smallest integer k in such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by χΣ″(G). In this paper, by using the famous Combinatorial Nullstellensatz, we determine χΣ″(G) for any planar graph G with Δ(G) ≥ 13.

Suggested Citation

  • Xu, Changqing & Li, Jianguo & Ge, Shan, 2018. "Neighbor sum distinguishing total chromatic number of planar graphs," Applied Mathematics and Computation, Elsevier, vol. 332(C), pages 189-196.
    • Handle: RePEc:eee:apmaco:v:332:y:2018:i:c:p:189-196
    DOI: 10.1016/j.amc.2018.03.013
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    References listed on IDEAS

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    1. Weifan Wang & Danjun Huang, 2014. "The adjacent vertex distinguishing total coloring of planar graphs," Journal of Combinatorial Optimization, Springer, vol. 27(2), pages 379-396, February.
    2. Jingjing Yao & Xiaowei Yu & Guanghui Wang & Changqing Xu, 2017. "Neighbor sum distinguishing total coloring of 2-degenerate graphs," Journal of Combinatorial Optimization, Springer, vol. 34(1), pages 64-70, July.
    3. Xiaohan Cheng & Guanghui Wang & Jianliang Wu, 2017. "The adjacent vertex distinguishing total chromatic numbers of planar graphs with $$\Delta =10$$ Δ = 10," Journal of Combinatorial Optimization, Springer, vol. 34(2), pages 383-397, August.
    4. Hongjie Song & Changqing Xu, 2017. "Neighbor sum distinguishing total coloring of planar graphs without 4-cycles," Journal of Combinatorial Optimization, Springer, vol. 34(4), pages 1147-1158, November.
    5. Cunquan Qu & Guanghui Wang & Guiying Yan & Xiaowei Yu, 2016. "Neighbor sum distinguishing total choosability of planar graphs," Journal of Combinatorial Optimization, Springer, vol. 32(3), pages 906-916, October.
    6. Hualong Li & Laihao Ding & Bingqiang Liu & Guanghui Wang, 2015. "Neighbor sum distinguishing total colorings of planar graphs," Journal of Combinatorial Optimization, Springer, vol. 30(3), pages 675-688, October.
    7. Yang, Donglei & Sun, Lin & Yu, Xiaowei & Wu, Jianliang & Zhou, Shan, 2017. "Neighbor sum distinguishing total chromatic number of planar graphs with maximum degree 10," Applied Mathematics and Computation, Elsevier, vol. 314(C), pages 456-468.
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    Cited by:

    1. Chao Song & Changqing Xu, 2020. "Neighbor sum distinguishing total colorings of IC-planar graphs with maximum degree 13," Journal of Combinatorial Optimization, Springer, vol. 39(1), pages 293-303, January.
    2. Anholcer, Marcin & Cichacz, Sylwia, 2019. "Note on the group edge irregularity strength of graphs," Applied Mathematics and Computation, Elsevier, vol. 350(C), pages 237-241.

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